For the sake of brevity, the remaining expansions will be given
only for a flat background. The complete expressions can be
found, e.g., in [1]}.
The simplest of the remaining relevant bilinear expansions is
\(\Big ( \int d^4 x \sqrt{-g^{\prime }}{R^{\prime }}^2\Big )^{(2)}_{\rm flat}&=&\int d^4 x \,\,\,h^{\mu \nu }\Big [\eta _{\alpha \beta }\eta _{\mu \nu }\operatorname{\square }^2- \eta _{\mu \nu }\operatorname{\square }\partial _\alpha \partial _\beta \nonumber \\&&- \,\, \eta _{\alpha \beta }\partial _\mu \partial _\nu \operatorname{\square }\,+ \,\partial _\mu \partial _\nu \partial _\alpha \partial _\beta \Big ] h^{\alpha \beta }.\) 
